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Random Schrödinger operators on long boxes, noise explosion and the GOE

Authors: Benedek Valkó and Bálint Virág
Journal: Trans. Amer. Math. Soc. 366 (2014), 3709-3728
MSC (2010): Primary 60B20, 81Q10
Published electronically: February 6, 2014
MathSciNet review: 3192614
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Abstract: It is conjectured that the eigenvalues of random Schrödinger operators at the localization transition in dimensions $ d\ge 2$ behave like the eigenvalues of the Gaussian Orthogonal Ensemble (GOE). We show that there are sequences of $ n\times m$ boxes with $ 1\ll m\ll n$ so that the eigenvalues in low disorder converge to Sine$ _1$, the limiting eigenvalue process of the GOE. For the GOE case, this is the first example where Wigner's famous prediction is proven rigorously: we exhibit a complex system whose eigenvalues behave like those of random matrices.

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Additional Information

Benedek Valkó
Affiliation: Department of Mathematics, University of Wisconsin-Madison, Madison,Wisconsin 53706

Bálint Virág
Affiliation: Department of Mathematics, University of Toronto, Ontario, Canada M5S 2E4

Received by editor(s): December 13, 2011
Received by editor(s) in revised form: October 2, 2012
Published electronically: February 6, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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